In the local-density approximation (LDA),
the many-electron problem is approximated by
a set of single-particle equations which are solved with the
self-consistent field method. The total energy is minimized.
The total energy is taken to be the sum of a kinetic energy, *T*,
the classical Hartree term for the electron density, ,
the electron-nucleus energy, , and the exchange-correlation
energy, , which takes into account approximately the fact that
an electron does not interact with itself, and that electron
correlation effects occur.

One solves the Kohn-Sham orbital equations

with

The charge density is given by

where the 2 accounts for double occupancy of each spatial orbital
because of spin degeneracy.
The potential, , is the external potential;
in the atomic case, this is where *Z* is the atomic number.
The exchange correlation potential, , is a function only of
the charge density, i.e., .
We use the functional of
Vosko, Wilk, and Nusair (1980)[4],
as described above.

The various parts of the total energy are given by:

and

where is the exchange-correlation energy per particle for the uniform electron gas of density . This approximation for is the principal approximation of the LDA.